On robust pricing and hedging and the resulting notions of weak arbitrage

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On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob lój University of Oxford obloj@maths.ox.ac.

1 On robust pricing and hedging and the resulting notions of weak arbitrage Jan Ob lój University of Oxford based on joint works with Alexander Cox (University of Bath) 5 th Oxford Princeton Workshop, Princeton, March 2009

2 Outline Principal Questions and Answers Financial Problem (2 questions) Methodology (2 answers) Double barrier options Introduction and types of barriers Double no touch example Theoretical framework and arbitrages Pricing operators and arbitrages No arbitrage vs existence of a model

3 Robust techniques in quantitative finance Oxford Man Institute of Quantitative Finance March 2010

4 Model risk: Robust methods: principal ideas Any given model is unlikely to capture the reality. Strategies which are sensitive to model assumptions or changes in parameters are questionable. We look for strategies which are robust w.r.t. departures from the modelling assumptions. Market input: We want to start by taking information from the market. E.g. prices of liquidly traded instruments should be treated as an input. We can then add modelling assumptions and try to see how these affect, for example, admissible prices and hedging techniques.

5 Model risk: Robust methods: principal ideas Any given model is unlikely to capture the reality. Strategies which are sensitive to model assumptions or changes in parameters are questionable. We look for strategies which are robust w.r.t. departures from the modelling assumptions. Market input: We want to start by taking information from the market. E.g. prices of liquidly traded instruments should be treated as an input. We can then add modelling assumptions and try to see how these affect, for example, admissible prices and hedging techniques.

6 Robust pricing and hedging: 2 questions The general setting and challenge is as follows: Observe prices of some liquid instruments which admit no arbitrage. ( interesting questions!) Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

7 Robust pricing and hedging: 2 questions The general setting and challenge is as follows: Observe prices of some liquid instruments which admit no arbitrage. ( interesting questions!) Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

8 Robust pricing and hedging: 2 questions The general setting and challenge is as follows: Observe prices of some liquid instruments which admit no arbitrage. ( interesting questions!) Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

9 Robust pricing and hedging: 2 questions The general setting and challenge is as follows: Observe prices of some liquid instruments which admit no arbitrage. ( interesting questions!) Q1: (very) robust pricing Given a new product, determine its feasible price, i.e. range of prices which do not introduce an arbitrage in the market. Q2: (very) robust hedging Furthermore, derive tight super-/sub- hedging strategies which always work. E.g.: Put-Call parity, Up-and-in put

10 Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? Suppose: (S t ) is a continuous martingale under P = Q, we see market prices C T (K) = E(S T K) +, K 0. Equivalently (S t : t T) is a UI martingale, S T µ, µ(dx) = C (x)dx. Via Dubins-Schwarz S t = B τt is a time-changed Brownian motion. Say we have O T = O(S) T = O(B) τt. We are led then to investigate the bounds LB = inf EO(B) τ, and UB = sup EO(B) τ, τ τ for all stopping times τ: B τ µ and (B t τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. The bounds are tight: the process S t := B τ t defines an T t asset model which matches the market data.

11 Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? Suppose: (S t ) is a continuous martingale under P = Q, we see market prices C T (K) = E(S T K) +, K 0. Equivalently (S t : t T) is a UI martingale, S T µ, µ(dx) = C (x)dx. Via Dubins-Schwarz S t = B τt is a time-changed Brownian motion. Say we have O T = O(S) T = O(B) τt. We are led then to investigate the bounds LB = inf EO(B) τ, and UB = sup EO(B) τ, τ τ for all stopping times τ: B τ µ and (B t τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. The bounds are tight: the process S t := B τ t defines an T t asset model which matches the market data.

12 Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? Suppose: (S t ) is a continuous martingale under P = Q, we see market prices C T (K) = E(S T K) +, K 0. Equivalently (S t : t T) is a UI martingale, S T µ, µ(dx) = C (x)dx. Via Dubins-Schwarz S t = B τt is a time-changed Brownian motion. Say we have O T = O(S) T = O(B) τt. We are led then to investigate the bounds LB = inf EO(B) τ, and UB = sup EO(B) τ, τ τ for all stopping times τ: B τ µ and (B t τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. The bounds are tight: the process S t := B τ t defines an T t asset model which matches the market data.

13 Q1 and the Skorokhod Embedding Problem Q1: What is the range of no-arbitrage prices of an option O T given prices of European calls? Suppose: (S t ) is a continuous martingale under P = Q, we see market prices C T (K) = E(S T K) +, K 0. Equivalently (S t : t T) is a UI martingale, S T µ, µ(dx) = C (x)dx. Via Dubins-Schwarz S t = B τt is a time-changed Brownian motion. Say we have O T = O(S) T = O(B) τt. We are led then to investigate the bounds LB = inf EO(B) τ, and UB = sup EO(B) τ, τ τ for all stopping times τ: B τ µ and (B t τ ) a UI martingale, i.e. for all solutions to the Skorokhod Embedding problem. The bounds are tight: the process S t := B τ t defines an T t asset model which matches the market data.

14 Q2 and pathwise inequalities Q2: if we see a price outside the bounds (LB,UB) can we (and how) realise a risk-less profit? Consider UB. The idea is to devise inequalities of the form O(B) t N t + F(B t ), t 0, with equality for some τ with B τ µ, and where is a martingale (i.e. trading strategy), EN τ = 0. Then UB = EF(S T ) and + F(S t ) is a valid superhedge. It involves dynamic trading and a static position in calls F(S T ). Furthermore, we want (N τt ) explicitly. We are naturally restricted to the family of martingales N t = N(B t,a t ), for some process (A t ) related to the option O t, e.g. maximum and minimum processes for barrier options.

15 Q2 and pathwise inequalities Q2: if we see a price outside the bounds (LB,UB) can we (and how) realise a risk-less profit? Consider UB. The idea is to devise inequalities of the form O(B) t N t + F(B t ), t 0, with equality for some τ with B τ µ, and where N t is a martingale (i.e. trading strategy), EN τ = 0. Then UB = EF(S T ) and N τt + F(S t ) is a valid superhedge. It involves dynamic trading (N τt ) and a static position in calls F(S T ). Furthermore, we want (N τt ) explicitly. We are naturally restricted to the family of martingales N t = N(B t,a t ), for some process (A t ) related to the option O t, e.g. maximum and minimum processes for barrier options.

16 Q2 and pathwise inequalities Q2: if we see a price outside the bounds (LB,UB) can we (and how) realise a risk-less profit? Consider UB. The idea is to devise inequalities of the form O(B) t N t + F(B t ), t 0, with equality for some τ with B τ µ, and where N t is a martingale (i.e. trading strategy), EN τ = 0. Then UB = EF(S T ) and N τt + F(S t ) is a valid superhedge. It involves dynamic trading (N τt ) and a static position in calls F(S T ). Furthermore, we want (N τt ) explicitly. We are naturally restricted to the family of martingales N t = N(B t,a t ), for some process (A t ) related to the option O t, e.g. maximum and minimum processes for barrier options.

17 Q2 and pathwise inequalities Q2: if we see a price outside the bounds (LB,UB) can we (and how) realise a risk-less profit? Consider UB. The idea is to devise inequalities of the form O(B) t N t + F(B t ), t 0, with equality for some τ with B τ µ, and where N t is a martingale (i.e. trading strategy), EN τ = 0. Then UB = EF(S T ) and N τt + F(S t ) is a valid superhedge. It involves dynamic trading (N τt ) and a static position in calls F(S T ). Furthermore, we want (N τt ) explicitly. We are naturally restricted to the family of martingales N t = N(B t,a t ), for some process (A t ) related to the option O t, e.g. maximum and minimum processes for barrier options.

18 Q2 and pathwise inequalities Q2: if we see a price outside the bounds (LB,UB) can we (and how) realise a risk-less profit? Consider UB. The idea is to devise inequalities of the form O(B) t N(B t,a t ) + F(B t ), t 0, with equality for some τ with B τ µ, and where N(B t,a t ) is a martingale (i.e. trading strategy), EN τ = 0. Then UB = EF(S T ) and N(S t,a S t ) + F(S t ) is a valid superhedge. It involves dynamic trading N(S t,a S t ) and a static position in calls F(S T ). Furthermore, we want (N τt ) explicitly. We are naturally restricted to the family of martingales N t = N(B t,a t ), for some process (A t ) related to the option O t, e.g. maximum and minimum processes for barrier options.

19 Scope of applications Answer to Q1 and pricing: in practice LB << UB, the bounds are too wide to be of any use for pricing. Answer to Q2 and hedging: say an agent sells O T for price p. She then can set up our super-hedge for UB. At the expiry she holds X = p UB + F(S T ) + N(S T,A S T ) O T. We have E Q X = 0 and X p UB. The hedge might have a considerable variance but the loss is bounded below (for all t T). The hedge is very robust as we make virtually no modelling assumptions and only use market input. This can be advantageous in presence of model uncertainty transaction costs illiquid markets. Numerical simulations indicate that a risk averse agent prefers robust hedges to delta/vega hedging.

20 Scope of applications Answer to Q1 and pricing: in practice LB << UB, the bounds are too wide to be of any use for pricing. Answer to Q2 and hedging: say an agent sells O T for price p. She then can set up our super-hedge for UB. At the expiry she holds X = p UB + F(S T ) + N(S T,A S T ) O T. We have E Q X = 0 and X p UB. The hedge might have a considerable variance but the loss is bounded below (for all t T). The hedge is very robust as we make virtually no modelling assumptions and only use market input. This can be advantageous in presence of model uncertainty transaction costs illiquid markets. Numerical simulations indicate that a risk averse agent prefers robust hedges to delta/vega hedging.

21 References and current works Previous works adapting the strategy: (lookback options) D. G. Hobson. Robust hedging of the lookback option. Finance Stoch., 2(4): , (one-sided barriers) H. Brown, D. Hobson, and L. C. G. Rogers. Robust hedging of barrier options. Math. Finance, 11(3): , (local-time related options) A. M. G. Cox, D. G. Hobson, and J. Ob lój. Pathwise inequalities for local time: applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab., 18(5): , As well as: forward starting options (D. Hobson and A. Neuberger,...) volatility derivatives (B. Dupire, R. Lee,...) double barrier options (A. Cox and J.O., arxiv: , ) variance swaps (M. Davis, J.O. and V. Raval)

22 Double barriers - introduction We want to apply the above methodology to derivatives with digital payoff conditional on the stock price reaching/not reaching two levels. Continuity of paths implies level crossings (i.e. payoffs) are not affected by time-changing. An example is given by a double touch: supt T S t b and inf t T S t b. 1 supu τ B u b and inf u τ B u b In general the option pays 1 on the event { ) ( ) and sup S t( b t T or inf t T S t ( ) } b

23 Double barriers - introduction We want to apply the above methodology to derivatives with digital payoff conditional on the stock price reaching/not reaching two levels. Continuity of paths implies level crossings (i.e. payoffs) are not affected by time-changing. An example is given by a double touch: supt T S t b and inf t T S t b. 1 supu τ B u b and inf u τ B u b In general the option pays 1 on the event { ) ( ) and sup S t( b t T or inf t T S t ( ) } b

24 Double barriers - introduction There are 8 possible digital double barrier options. However using complements and symmetry, it suffices to consider 3 types: double touch option new solutions to the SEP. double touch/no-touch option new solutions to the SEP. double no-touch option maximised by Perkins construction and minimised by the tilted-jacka (A. Cox) construction.

25 Double barriers - introduction There are 8 possible digital double barrier options. However using complements and symmetry, it suffices to consider 3 types: double touch option new solutions to the SEP. double touch/no-touch option new solutions to the SEP. double no-touch option maximised by Perkins construction and minimised by the tilted-jacka (A. Cox) construction.

26 Double barriers - introduction There are 8 possible digital double barrier options. However using complements and symmetry, it suffices to consider 3 types: double touch option new solutions to the SEP. double touch/no-touch option new solutions to the SEP. double no-touch option maximised by Perkins construction and minimised by the tilted-jacka (A. Cox) construction.

27 Double no touch: Answer to Q1 Write B t = sup s t B s, B t = inf s t B s : B t γ + ( B t ) inf{t : B t (γ (B t ),γ + ( B t ))} Maximises: P(B τ b and B τ b) γ (B t ) Perkins (1985) B t

28 Double no touch: Answer to Q1 Write B t = sup s t B s, B t = inf s t B s : B t γ + ( B t ) inf{t : B t (γ (B t ),γ + ( B t ))} Maximises: P(B τ b and B τ b) γ (B t ) Perkins (1985) B t

29 Double no touch: Answer to Q2 Consider pathwise inequality: 1 {ST b,s T b} 1 S T >b (b S T) + where b < S 0 < K. When S T > b, we get: K b + (S T K) + S T b K b K b 1 S T b 1 1 ST >b + (S T K) + K b b K

30 Double no touch: Answer to Q2 Consider pathwise inequality: 1 {ST b,s T b} 1 S T >b (b S T) + where b < S 0 < K. When S T b, we get: K b + (S T K) + S T b K b K b 1 S T b 0 (K S T) + 1 K b {ST >b} b K

31 Double no touch: Answer to Q2 Consider pathwise inequality: 1 {ST b,s T b} 1 S T >b (b S T) + where b < S 0 < K. K b + (S T K) + S T b K b K b 1 S T b This is a model free superhedging strategy for any b < K.

32 Double no touch: Answer to Q2 Consider pathwise inequality: 1 {ST b,s T b} 1 S T >b }{{} Digital call (b S T) + K b } {{ } Puts S T b K b 1 S T b }{{} Forwards upon hitting b + (S T K) + K b }{{} Calls =: H II (K) This is a model free superhedging strategy for any b < K, assuming (S t ) does not jump across the barrier b.

33 Double no touch: Answer to Q2 Consider pathwise inequality: 1 {ST b,s T b} 1 S T >b }{{} Digital call (b S T) + K b } {{ } Puts S T b K b 1 S T b }{{} Forwards upon hitting b + (S T K) + K b }{{} Calls =: H II (K) We would like to show that it is a hedging strategy in some model. It turns out that the above construction is not always optimal there are two more strategies H I, H III (K) we need to consider. Above we superhedged 1 ST >b as in Brown, Hobson, Rogers (2001) and it s good only for b < S 0 << b.

34 Double touch: superhedging Write P for the pricing operator. No arbitrage should imply: } P1 {ST b,s T b} {PH inf I, PH II (K 2 ), PH III (K 3 ) =: UB ( ) where the infimum is taken over values of K 2 > b, K 3 < b. Theorem ( Meta-Theorem ) No arbitrage iff ( ) holds and for any given curve of call prices there exists a stock price process for which ( ) is the price of the double no touch option.

35 Double touch: superhedging Write P for the pricing operator. No arbitrage should imply: } P1 {ST b,s T b} {PH inf I, PH II (K 2 ), PH III (K 3 ) =: UB ( ) where the infimum is taken over values of K 2 > b, K 3 < b. Theorem ( Meta-Theorem ) No arbitrage iff ( ) holds and for any given curve of call prices there exists a stock price process for which ( ) is the price of the double no touch option.

36 General setup We assume (S t : t T) takes values in some functional space P, and (S t ) has zero cost of carry (e.g. interest rates are zero). The set of traded assets X is given. On this set we have a pricing operator P which acts linearly on X, P : Lin(X) R. We say that there exists a (P, X) market model if there is a model (Ω, F,(F t ), Q,(S t )) with PX = E Q X, X X. We would like to have P admits no arbitrage on X there exists a market model Then we want to consider X {O T } for an exotic O T : P R and say P admits no arbitrage on X {O T } LB PO T UB there exists a (P, X {O T }) market model

37 General setup We assume (S t : t T) takes values in some functional space P, and (S t ) has zero cost of carry (e.g. interest rates are zero). The set of traded assets X is given. On this set we have a pricing operator P which acts linearly on X, P : Lin(X) R. We say that there exists a (P, X) market model if there is a model (Ω, F,(F t ), Q,(S t )) with PX = E Q X, X X. We would like to have P admits no arbitrage on X there exists a market model Then we want to consider X {O T } for an exotic O T : P R and say P admits no arbitrage on X {O T } LB PO T UB there exists a (P, X {O T }) market model

38 General setup We assume (S t : t T) takes values in some functional space P, and (S t ) has zero cost of carry (e.g. interest rates are zero). The set of traded assets X is given. On this set we have a pricing operator P which acts linearly on X, P : Lin(X) R. We say that there exists a (P, X) market model if there is a model (Ω, F,(F t ), Q,(S t )) with PX = E Q X, X X. We would like to have P admits no arbitrage on X there exists a market model Then we want to consider X {O T } for an exotic O T : P R and say P admits no arbitrage on X {O T } LB PO T UB there exists a (P, X {O T }) market model

39 Three notions of arbitrage Definition (Model free arbitrage) We say that P admits a model free arbitrage on X if there exists X Lin(X) with X 0 and PX < 0.

40 Three notions of arbitrage Definition (Model free arbitrage) We say that P admits a model free arbitrage on X if there exists X Lin(X) with X 0 and PX < 0. This coarsest notion is typically sufficient to derive no arbitrage bounds but not sufficient to give existence of a market model. Consider X = {(S T K) + : K K = {K 1,...,K n }}. No MFA implies interpolation of C(K) := P(S T K) + is convex and non-increasing. We could have C(K n 1 ) = C(K n ) > 0. But this leads to arbitrage strategies: if I have a model with S T K n a.s., I sell call with strike K n, if I have a model with P(S T > K n ) > 0 I sell call with strike K n and buy call with strike K n 1.

41 Three notions of arbitrage Definition (Model free arbitrage) We say that P admits a model free arbitrage on X if there exists X Lin(X) with X 0 and PX < 0. Definition (Weak arbitrage (Davis & Hobson 2007)) We say that P admits a weak arbitrage on X if for any model, there exists X Lin(X) with PX 0 but P(X 0) = 1, P(X > 0) > 0. Definition (Weak free lunch with vanishing risk) We say that P admits a weak free lunch with vanishing risk on X if there exists X n,z Lin(X) such that X n X (pointwise on P), X n Z, X 0 and lim PX n < 0.

42 Three notions of arbitrage Definition (Model free arbitrage) We say that P admits a model free arbitrage on X if there exists X Lin(X) with X 0 and PX < 0. Definition (Weak arbitrage (Davis & Hobson 2007)) We say that P admits a weak arbitrage on X if for any model, there exists X Lin(X) with PX 0 but P(X 0) = 1, P(X > 0) > 0. Definition (Weak free lunch with vanishing risk) We say that P admits a weak free lunch with vanishing risk on X if there exists X n,z Lin(X) such that X n X (pointwise on P), X n Z, X 0 and lim PX n < 0.

43 Call prices and no arbitrages Proposition (Davis and Hobson (2007)) Let X = {1,(S T K) + : K K} be finite. Then P admits no WA on X if and only if there exists a (P, X)-market model.

44 Proposition Call prices and no arbitrages Let X = {1,(S T K) + : K 0}. Then P admits no WFLVR on X if and only if there exists a (P, X)-market model, which happens if and only if C(K) = P(S T K) + 0 is convex and non-increasing, and C(0) = S 0, C +(0) 1, (1) C(K) 0 as K. (2) In comparison, P admits no model-free arbitrage on X if and only if (1) holds. In consequence, when (1) holds but (2) fails P admits no model-free arbitrage but a market model does not exist.

45 Call and digital call prices and no arbitrages Proposition Let X = {1,1 ST >b,1 ST b,(s T K) + : K 0}. Then P admits no WFLVR on X if and only if there exists a (P, X)-market model, which happens if and only if C(K) is as previously and P1 ST >b = C (b+) and P1 ST b = C (b ).

46 Call and digital call prices and no arbitrages Proposition Let X = {1,1 ST >b,1 ST b,(s T K) + : K 0}. Then P admits no WFLVR on X if and only if there exists a (P, X)-market model, which happens if and only if C(K) is as previously and P1 ST >b = C (b+) and P1 ST b = C (b ). Proposition Let X = {1,1 ST >b,1 ST b,(s T K) + : K K} be finite. Then P admits no WA on X if and only if there exists a (P, X)-market model. In both cases no WFLVR or no WA are strictly stronger than no model free arbitrage.

47 Call and digital call prices and no arbitrages Proposition Let X = {1,1 ST >b,1 ST b,(s T K) + : K 0}. Then P admits no WFLVR on X if and only if there exists a (P, X)-market model, which happens if and only if C(K) is as previously and P1 ST >b = C (b+) and P1 ST b = C (b ). Proposition Let X = {1,1 ST >b,1 ST b,(s T K) + : K K} be finite. Then P admits no WA on X if and only if there exists a (P, X)-market model. In both cases no WFLVR or no WA are strictly stronger than no model free arbitrage.

48 All our main results for digital double barriers are of this type with WA replacing WLVR for the case of finite family of traded strikes. Theorem Double barriers and no arbitrage Let P = C([0,T]). Suppose P admits no WFLVR on X = {forwards} {1,1 ST >b,1 ST b,(s T K) + : K 0}. Then the following are equivalent P admits no WFLVR on X {1 {ST b,s T b} }, there exists a (P, X {1 {ST b,s T b} }) market model, } P(1 {ST b, S T b} {P(H ) inf I ), P(H II (K 2 )), P(H III (K 3 )), } P(1 {ST b, S T b} {P(H ) sup I ), P(H II (K 1,K 2 )). (and we specify the hedges & strike(s) which attain inf/sup).

49 Summary Given a set of traded assets we want to construct robust super- and sub- hedging strategies of an exotic option. Further, we want them to be optimal in the sense that there exists a model, matching the market input, in which they are the hedging strategies. We carry out this programme for all types of digital double barrier options when the set of traded assets includes calls, digital calls and forward transactions. We introduce a formalism for the model free setup and define stronger notions of arbitrage (WFLVR and WA). There exists a market model (matching the input) iff appropriate no arbitrage holds. Further, the same holds if we add a double barrier, and this is equivalent to its price being within the bounds we derive.

50 Summary Given a set of traded assets we want to construct robust super- and sub- hedging strategies of an exotic option. Further, we want them to be optimal in the sense that there exists a model, matching the market input, in which they are the hedging strategies. We carry out this programme for all types of digital double barrier options when the set of traded assets includes calls, digital calls and forward transactions. We introduce a formalism for the model free setup and define stronger notions of arbitrage (WFLVR and WA). There exists a market model (matching the input) iff appropriate no arbitrage holds. Further, the same holds if we add a double barrier, and this is equivalent to its price being within the bounds we derive.

51 Summary Given a set of traded assets we want to construct robust super- and sub- hedging strategies of an exotic option. Further, we want them to be optimal in the sense that there exists a model, matching the market input, in which they are the hedging strategies. We carry out this programme for all types of digital double barrier options when the set of traded assets includes calls, digital calls and forward transactions. We introduce a formalism for the model free setup and define stronger notions of arbitrage (WFLVR and WA). There exists a market model (matching the input) iff appropriate no arbitrage holds. Further, the same holds if we add a double barrier, and this is equivalent to its price being within the bounds we derive.

52 Summary Given a set of traded assets we want to construct robust super- and sub- hedging strategies of an exotic option. Further, we want them to be optimal in the sense that there exists a model, matching the market input, in which they are the hedging strategies. We carry out this programme for all types of digital double barrier options when the set of traded assets includes calls, digital calls and forward transactions. We introduce a formalism for the model free setup and define stronger notions of arbitrage (WFLVR and WA). There exists a market model (matching the input) iff appropriate no arbitrage holds. Further, the same holds if we add a double barrier, and this is equivalent to its price being within the bounds we derive.

Model-independent bounds for Asian options

Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique University of Michigan, 2nd December,